2-4-1 Subtraction of Real Numbers
Rule: To subtract ADD the OPPOSITE.
This sounds easy.
Think about a problem 3 – 6. The operation in this problem is subtraction. The 6 is positive.
To follow the rule change the subtraction operation to addition and
the positive 6 to a negative 6.
The result is 3 +(-6). Now follow the addition rules to get a result of –3.
It may be easier to think UP 3 DOWN 6 from the original 3-6 to arrive at –3.
The word minus can be used as a verb or an adjective in an equation.
It can mean to subtract or denote a negative number.
Example: -5.6-(-3) ADD the OPPOSITE –5.6+(3) = -2.6
Notice the subtraction operation changed to addition, and the opposite of –3 becomes 3.
- 3.2 – 2.5 ADD the OPPOSITE –3.2 + (-2.5) =-5.7
The – in front of the 2.5 is a verb. It is a subtraction and this is what is changed to addition.
The – in front of the 3.2 is an adjective. It tells what kind of number 3.2 is.
An easy way to do these problems without writing down the new addition problem is think “UP” and “DOWN”
at the appropriate time.
-2/3 –1/4 Think: Down 2/3 down 1/4 more. Get a common denominator.
8/12 down and 3/12 down is –11/12.
3-8 Think: UP 3 then Down 8 for 5 in the hole. 3-8 = -5
-81-12 Think Down 81 then Down 12 more.
–81 –12 = -93.
When two negative are next to each other, you must write the new addition problem on paper.
5-(-3)
ADD the OPPOSITE 5 + (+3) Notice the use of the rule. 5-(-3) = 5+3 = 8
-8 –(-4) ADD the OPPOSITE –8 + (+4). Now think Down 8 then up 4.
-8-(-4) = -8+4 = -4
When double negatives are in long lines of addition and subtraction, you must write the new addition.
However, you only need to change the double negative.
Example: -7-9-(-4)-3+9-(-4) Changes on paper to –7-9+4-3+9+4.
Then think DOWN 7, Down 9 more, then UP 4, Down 3, UP 9, and UP 4. The result is –2.
Practice:
a) 5-8
81-43
-23-24
-34-(-21)
-3-(-8)
-8x-12x
b)
-3x-4x-2x+x
c)
-d-(-3d)+7d-3d-(-5d)
d)
w  2w  3w 5w



3  3  4
6
-7e+3e-10e+3e-2e-2e
2c-3b+5c-3c-(-3b)
-6a-(-3a)-(-2a)-8a+2a-12a-6a
-4r-(-3r)-9r-3r+2r+r-6s+8s
1
5
 2  3
4 a   5 b   b  a  3a  7b
3
3
4
6


2-4-2 Integer Arithmetic
Always
Neg - Pos = Neg
(add)
a)
-8 - 13 = -21
b)
-7 - 15 =
c)
-81 - 130 =
d)
-28m - 313m =
(Watch, because some may not be like terms.)
Depends on which is the larger absolute value.
Pos - Neg = Pos (add) Pos – Pos (subtract) Pos – Pos (subtract) Neg- Neg (subtract) Neg – Neg
(subtract)
13 - (-15) = 28
5 - 8 = -3
8-5=3
-8 - (-3) = -5
-3 - (-8) = 5
123 - (-15) =
25 - 82 =
81 - 5 =
-108 - (-13) =
-31 - (-81) =
43 - (-18) =
3-8=
12 - 7.5 =
-58 - (-35) =
-53 - (-108) =
130n - (-105n) =
21b - 58b =
48k - 35k =
-18j - (-13j) =
-p - (-12p) =
e)
-3a-7a=
54y - (-203y) =
b - 8b =
4q - q =
-j - (-j) =
-0.1u - (-1.2u) =
f)
-5.32e - 0.236e=
5.4e - (-7.3e) =
0.21m - 0.58 =
4.8k - 3.5k =
-8j - (-1.3j) =
-0.3p - (-0.82p) =
g)
h)
-5.2a2 - 0.36a=
-0.32q - 0.2536q=
0.09e - (-2e) =
0.32q - (-0.2536q)=
0.05m - 0.5m =
0.12q - 0.2536q=
4k - 0.088k =
0.2536q - 0.32q =
-0.78j - (-0.3j) =
-0.3 - (-7.12k) =
-0.32q – (-0.2536)q= -0.2536q – (-0.32q)
i)
2
1
3  2 =
5
4
2  1
  2  =
5  4
2
1
2 =
5
4
2
1
3 2 =
5
4
2  1
3   2  =
5  4
2  1
3   7  =
5  4
j)
2
1
 s  s=
3
4
2x  x 
  =
3  4
2
3
s  s=
3
4
2
1
s  s=
3
4
2  1 
 s  s =
3  4 
2
 3 
 y  y=
3
 4 
k)
5
5
3
y y=
12
4
5
5
 3y 
y   =
12
 4 
5
5
3
y 8 y=
12
4
5
5
3
y y=
12
4
5
5
 3 
y  y
12
 4 
5
5
3 

x   7 x 
12
4 

l)
2 1
8 t  t =
5 4
2
1 

t   8 t  =
5
4 

3
3
t 5 t 
8
4
2 1
8 t  t=
5 4
2  1 
8 t    t  =
5  4 
2  3 
8 t   8 t  =
5  4 
m)
a a
  =
3 4
a  a
 =
3  4
a 3a
 =
3 4
a a
 =
3 4
a  a
  =
3  4
a  a
  =
4  4
2-4-3 More Practice
1
(-85) - 6 - 69 - 10 =_______________
2
(-36) - 21 - 11 - (-47) =_______________
3
(-25) - 14 - 96 - (-14) =_______________
4
(-50) - (-68) - (-12) - (-92) - (-68) - (-89) =_______________
5
(-83) - 13 - 91 - (-27) - (-28) - (-9) =_______________
6
(2.11)-(-4.14)-(6.29)-(19.53)-(19.16)-(-2.92) =_______________
7
(-15.58)-(-12.02)-(18.64)-(-15.47)-(-11.5)-(-2.87) =_______________
8
90 - 15 - (-64) + (-69) =_______________
9
86 + 41 - (-88) + (-6) =_______________
10
56 + 95 - (-96) - 65 =_______________
11
93 + (-30) - 29 + 30 - 86 - 49 =_______________
12
(-67) - (-17) - 74 - 96 - (-6) - 38 =_______________
13
4 + (-9) + (-82) - (-3) - (-13) + 63 =_______________
14
(-66) + (-15) - (-47) + 55 + 3 + (-28) =_______________
15
(-15.57)-(2.22) + (18.4) + (6.5)-(-9.63)-(18.7) =_______________
16
(-60) - 49 - 86 + (-72) - (-77) + (-70) =_______________
17
(-40) +(-92) - 34 -(-21) + 33 +(-90) +(-65) -(-62) =_______________
18
(-1) - (-62) + (-41) - 82 + 31 - 63 - (-17) - 26 =_______________
19
(-5) - 72 - 32 -(-1) + 68 - 85 + 33 -(-61) +(-42) - 86 =_______________
20
(-9 1/ 3)-( 2/ 3)+( 1/ 4)-(-2 1/ 5)+(-9 1/ 2)=_______________
21
(-2 1/ 2)-( 2 1/ 3)+(-4 1/ 3)+(-7 1/ 2)=_______________
22
(-2 2/ 3)+( 8 7/ 8)+(-5 7/ 8)+( 7 3/ 4)=_______________
23
( 4 1/ 2)+( 8 1/ 5)+( 4 1/ 10)-(-7 3/ 5)=_______________
2-4-4 Word Problems
You may use a calculator on any problem that requires multiplication or division.
a) Tiny Car company made 5.6 million last year.
This year they made 12.3 million. How much
more did Tiny Car company make this year?
A tree was 13.4 meters tall in 1999. In 2002 it
grew to 14.1 meters.
How much did the tree grow?
b) There are 5 people on the elevator. Together
they weigh 925.98 lbs. Find the average weight.
What is the weight of 36 cubic feet of water? A
cubic foot of water weighs 62.5 pounds.
c) Kathy worked 5.4 hours of overtime at $25.42
per hour. How much did she make?
Jan put three packages on a scale. Individually
they weigh 4.2 kilograms, 2.37 kilograms, and .45
kilograms. What is on the scale display?
d) A wool scarf that was 3.2 feet long before the
wash is now 2.8 feet long. How much did the
scarf shrink?
If the price of a gallon of gasoline rises from $0.98
to $1.34, how much does the price of a gallon
rise?
e) David had the flue for Thanksgiving weekend.
His temperature was 103.2 at the highest. His
normal temperature is 98.6. How much higher
was his temperature during his fever?
There are 2.54 centimeters in one inch. How
many inches are there in 51.78 centimeters?
f) What was its average speed in miles per hour if
a plane flew 1856.4 miles in 5.2 hours?
3.2 pounds of potatoes, 2.5 pounds of apples, .75
pound of onions are bought for stew. What is the
total weight?
g) At $6.42 a cubic foot, how much do 8.25 cubic
feet of concrete cost?
What is the average number of miles Daniel drove
on a 1345 mile trip done in 6 ½ days? (hint
change the ½ to .5.)
h) Women live, on average to 73.5 years old.
Men live to 68.9 years old. How much longer do
women live on average?
i) 1 inch = 2.54 centimeters. How many
centimeters are there in 56 inches?
There are 1.6 kilometers in a mile. How many
miles are there in 98.7 kilometers?
Gerry makes $7.58 an hour. He grossed $306.99
last week. How many hours did he work?
j) How many meters are there in 567 yards?
1yard = 0.914 meters.
In 1990 there were 359.9 thousand people in
Mytown. By 1996 there were 8.6 thousand more
people town. How many people lived in Mytown in
1996?
k) If 5.3 pounds of nails cost $7.65 what is the
price of one pound of nails?
Three parts are to be welded. They are 3.45 in.,
7.8 in., and .98 in. in length. What will be the
final length?
l) 2.5 yards of polar fleece cost $26 how much
does one yard cost?
Last year the precipitation was 3.5 inches in
December, 2.5 inches in January and 3 inches in
February. What was the average precipitation for
last winter?
m) The television costs $0.08 per hour to run.
How much will it cost to watch 3.45 hours?
The trip to the outlet store is 5.3 miles. From
there we will go to the campus library that is 3.02
miles. From the library it is a 2 mile straight ride
home. How far will we drive?
n) Ammon makes 6.45 per hour. How worked 5.3
hours Monday, 8 hours on Tuesday, 2.75 hours on
Wednesday, .75 hours on Thursday, and 8 hours
on Friday. How much did he make that week?
Randy drives 34.5 miles to grandma's house. His
brother, Mike, drives 72.3 miles. How much
farther is does Mike drive?
o) Lacey has a piece of wire that is 3.5 feet long.
She needs to cut 4 pieces off that are .85 feet
each. How much wire is left?
Last time I filled the tank, my mileage was
1,345.8. Now it reads 1383.2. How far did I drive
on the last tank of gas? If my tank hold 15
gallons, what was the miles per gallon?
p) The baby weighs 49 pounds. The weight is to
be charted in kilograms. What would you chart?
One pound is equal to 0.45 kilograms.
Tony sold 3.45 shares of one stock, 23.4 shares of
another and 15 shares of a third stock. How many
shares did he sell?
q) One mile is equal to 1.6 kilometers. How many
kilometers are there in 73 miles?
The average speed of the Enterprise is 521 light
years per hour. How far did the ship fly in 0.325
hours.?